[Proof of Theorem REF ]
Since the weak-Hilbert property is a birational invariant among smooth proper varieties (Proposition REF ), we may and do assume that \(X\)  has trivial tangent bundle. In particular, since \(k\)  is of characteristic zero, there is a finite field extension \(L/k\)  such that \(X_L\)  is an abelian variety (see [1]}). Replacing \(L\)  by a finite field extension if necessary, by the potential density of rational points on abelian varieties (see [2]}), we may assume that \(X(L)\)  is Zariski-dense in \(X\) . It then follows from Theorem REF  that \(X_L\)  has the weak-Hilbert property over \(L\) . Finally, it is shown in [3]} that \(X_{\overline{k}}\)  is special. This concludes the proof.
